parameter optimization in wave energy design by a genetic...

The 32nd International Workshop on Water Waves and Floating Bodies, Dalian, China, 23-26 April, 2017

Parameter Optimization in Wave Energy Design by a GeneticAlgorithm

Marianna Giassi∗, Malin Goteman

Department of Engineering Sciences, Uppsala University, Uppsala, SwedenEmail address of presenting author: [email protected]

HIGHLIGHTS

• A tool for optimization of parameters of a single wave energy converter using a genetic algorithmwas developed and validated against a parameter sweep optimization of the variables.

• The procedure combines the genetic algorithm with a multiple scattering method for the calcu-lation of the hydrodynamics (Goteman M. et al., 2015), providing fast computational time.

• The method was then modified and applied to optimization of the spatial layout of a wave powerarray of identical devices, that interact hydrodynamically by scattered and radiated waves. Theresulting optimal park layouts avoid destructive interactions and get a q-factor slightly above 1.

1 INTRODUCTION

Wave energy conversion is one of the promising renewable energy technologies that is now facing thechallenging steps towards early commercialization. Many of the technologies developed consist ofsmall devices that are designed to be deployed in large parks of many units to produce a considerableamount of power. The wave energy converter (WEC) developed at Uppsala University belongs tothis group of technologies; it consists of a floating buoy connected to a linear direct-driven permanentmagnet generator on the seabed (Leijon M. et al., 2009). When deployed, the devices interact witheach other, both hydrodynamically by scattered and radiated waves, and electrically, leading to anincrease or decrease of the power production depending on many parameters such as park layout,number of devices, separating distance, wave direction, etc.

This paper focuses on the development and implementation of a tool able to optimize some ofthe principal parameters (buoy’s radius, buoy’s draft and damping coefficient of the generator) of asingle point absorber WEC using a genetic algorithm (GA). After validation of the method againstparameter sweep (PS), the tool has been improved to optimize the spatial layout of a park of WECs.

Research in wave energy park optimization by means of a genetic algorithm has been carried outby Child B.F.M. & Venugopal V. (2010), Child B.F.M. et al. (2011) and Sharp C. & DuPont B. (2016).Child B.F.M. & Venugopal V. (2010) used parabolic intersection and genetic algorithm with differentkind of tuning of the devices and optimized the spatial configuration of 5 identical devices upon poweroutput of the array; Sharp C. & DuPont B. created a model to determine array configurations andincluded both power output and costs in the evaluation function, both with binary and continuousGA. Child B.F.M. et al. (2011) included optimization of power take off characteristics given a fixedlayout and optimization of array layout of 10 identical WECs, given fixed power take off coefficients.

2 METHOD

A genetic algorithm is one method among the more generic evolutionary algorithms which are based onthe theory of biological evolution. In this kind of routines, the optimization depends on an “evolution”of a set of solutions, “genetically developing” every iteration towards the optimum. These methodsrely on an intelligent search of a large but finite solution space using statistical methods and can dealwith discrete variables and non continuous cost functions (Haupt R.L. & Haupt S.E., 2004).

After definition of the GA parameters (see Table 1), a first population is generated by randomlysampling from a pool of defined values; this is the first set of chromosome (nPop), each of them


Fig. 1: Genetic algorithm optimal solutions vs parameter sweep values and optimum. Surfaces showthe PS results for a given fixed (optimal) value of the draft (a), the radius (b) and the γ coefficient(d). Diamonds represent GA solutions. (c) and (d) show a zoom of the GA results (diamonds) withparameter sweep optimum (cross).

consisting in a number (nVar) of different genes (nGene). Then, the first population is evaluated andranked, and the convergence criteria are checked. The following convergence criteria were implementedin the code: 1) a maximum number of iterations (MaxIt) is reached; 2) all the chromosomes inthe actual population are the same; 3) the solution ceases to improve after a certain number (cutoff ) of iterations. As long as convergence is not reached, reproduction is performed, where some ofthe chromosomes chosen as parents will be coupled to generate a new part of the population calledoffspring. Reproduction is executed in several steps: natural selection, pairing, mating or crossover,mutation.

In the presented study two slightly different GAs were implemented in MATLAB coupled withan analytical fast multiple scattering method for the calculation of the hydrodynamic parameters ofthe fully hydrodynamically coupled WECs developed by Goteman M. et al. (2015). The input to themodel consists of time series of irregular waves measured off-shore at the west coast of Sweden.

2.1 Genetic algorithm for a single device

The goal of the single WEC genetic algorithm tool is to find the optimal value of the radius (R) anddraft (d) of the buoy and the damping coefficient (γ) of the generator, considering the point-absorberWEC described above; consequently, the number of different genes (nGene) in every chromosome isthree: R, d, γ. The evaluation function computes the total mean absorbed power of the WEC overan hour of time; the negative of this value will then be minimized to find the optimum solution. Theparameter space is discretized so that solution values are placed on a grid; the resolution and widenessof this grid is customizable and depends on the range of initial set parameters.

Natural selection determines which percentage (selection rate) of the individuals will survive and


Table 1: Genetic algorithm parameters settings.

Parameter Description 1 WEC Array

Nb Number of WEC 1 4,5,7,9nPop Initial population size (number of chromosomes) 16 12nGene Number of different genes 3 1Nvar Number of genes in every chromosome 3 Nb·nGeneMaxIt Maximum number of iteration/generation 100 250selection rate Fraction of nPop that is selected for mating 0.5 0.5mutation rate % of population to be mutated 0.2 - 0.3 0.2elitism rate Number of best solutions kept unaltered in the next generation 1 3cut off Iteration needed to converge if solution doesn’t improve 25 −

continue to the next generation; this fraction of the chromosomes is selected after the population hasbeen ranked from the best to the worst objective function value. Afterwards, the chromosomes in thesurvived population are paired in order to mate and reproduce two offsprings. During crossover, onegene is randomly chosen as “crossover point” and then swapped between the two parents chromosomes,so that distinctive genes from both individuals are inherited by every child. Finally, new geneticmaterial is introduced by mutation, which randomly changes a chosen percentage of variables in thepopulation. Elitism keeps the first set percentage of the ranked population unchanged in the followinggeneration, to preserve the best solutions from potentially negative mutations.

2.2 Extension of the method for an array

After validation, the algorithm has been modified to perform layout optimization of a park of manyunits. This enables multiple parameter optimization in wave energy parks, where the parameter spaceis so large so that a parameter sweep optimization becomes computationally unfeasible. The singledevices are allowed to take their coordinates inside a fixed gridded ocean area. In this case the numberof different genes (nGene) is equal to one (couple of coordinates [xi, yi], where i is the i-th device in thepark). Therefore, there will be Nb ·nGen genes in every chromosome (Nvar), where Nb is the numberof WECs in the array. Natural selection and mating are performed in the same way as describedfor the single WEC. Crossover is performed by choosing a random variable as “crossover point” thatdivides the chromosome in two parts; these are then swapped between the two parents individuals.

3 RESULTS

3.1 Single device parameter optimization

In the single device GA optimization performed, the parameters are allowed to assume the followingrange of values: R = 1 : 0.5 : 5 m, d = 0.2 : 0.05 : 0.4 m and γ = 15 : 1 : 2000 kNs/m. The model inputis a time series of irregular waves characterized by significant wave height Hs = 1.53 m and energyperiod Te = 5.01 s. The maximum number of iterations is set to 100 and the cut off iteration equalto 25. Mutation rate is 20%. 20 different simulations have been performed and validated againstparameter sweep of the variables (i.e. computation of the power output for every combination ofradius, draft and γ coefficient possible); the results of the validation are shown in Fig. 1.

The agreement between the PS computation and the GA simulations is very good: all the solutionsfound by the GA (colored diamonds) are located in the region of maximum power output calculatedby parameters sweep (surfaces); the differences between the PS optimum value and the GA resultshas been, in all 20 simulations, less than 0.2% regarding the final average power output of the WEC.Optimal radius and draft were found in 100% of the simulations and the difference between γ coeffi-cients found by the GA and the optimum PS value is less than 8%. This is most likely related to thenumber of iterations that the tool was allowed to perform.

Even though the GA model of the single WEC is simple, the saving in computational time isimportant. One run of PS with the previous mentioned parameter space took 87.2 min, while 85%


Fig. 2: Results of GA optimization after 250 iterations.

of the GA runs took less than 10 min to reach convergence. Is worth noticing that, in almost allthe cases, convergence of type 3 was reached, meaning that the actual solution was found a cut offnumber of iterations (in this case 25) before maximum number of iterations (100). Simulations wereperformed on a desktop PC with 16 parallel Intel(R) Xeon(R) 2.40 GHz processors and 32 GB RAM.

3.2 Wave energy arrays layout optimization

Simulations of arrays of 4-9 devices were performed; each WEC has the same design characteristics(R = 3 m, d = 0.45 m and γ = 140 kNs/m). The ocean area is 2500 m2, gridded every 10 m bothin x and y directions; devices are allowed to take coordinates on this grid. Unidirectional irregularwaves are moving along the x axis. Fig. 2 shows layout results obtained for different park sizes, after250 GA iterations. As expected, the best configuration for 4 and 5 WECs in such a large area is theperfect alignment of the park with the incoming wave front. When the ocean area and the separationdistance do not allow a complete alignment (i.e. for 7 and 9 WECs), the optimal power is produced byadding to the complete line a supplemental front line with buoys located every second spot. The powerproduction found by the GA has an increase of the output between 3% and 7%, depending on thenumber of WECs; the q-factor for the resulting layouts is slightly above 1, meaning that destructivehydrodynamical interaction is avoided and slight positive interaction is gained.

4 CONCLUSIONS

The work presented here has shown the potentiality of GA implementation for wave energy design:optimization method for single WEC’s parameters gave accurate results in little computational time,compared to PS. Moreover, the results have been successfully validated. Then, given specified pa-rameters of a wave energy converter’s array (i.e. size of the WECs and power take-off characteristics,ocean area and distance required between the devices), the genetic algorithm implemented was able tofind a spatial configuration which avoid destructive hydrodynamical interactions (q-factor of the arrayslightly > 1). The solutions are in some sense independent from the number of WECs and consist of aregular geometrical pattern. This would enable a fast pre-evaluation of the hydrodynamic interactionsbefore the array’s deployment process starts. Moreover, different runs of the GA tool with a sufficientnumber of iterations have led to the same configurations results.

REFERENCES

Child B.F.M, Venugopal V., 2010. Optimal configuration of wave energy device arrays. Ocean Engineering, 37, 1402-1417.Child B.F.M., Cruz J., Livingstone M., 2011. The development of a tool for optimizing arrays of wave energy converters,Proc. of 9th EWTEC, Southampton, UK.Goteman M. et al., 2015. Fast modeling of large wave energy farms using interaction distance cut-off. Energies, 8,13741-13757.Haupt R.L., Haupt S.E., 2004. Practical Genetic Algorithms, 2nd Edition. ISBN 0-471-45565-2, John Wiley & Sons, Inc.Leijon M. et al., 2009. Catch the wave to electricity: the conversion of wave motions to electricity using a grid-orientedapproach. IEEE Power & Energy Mag., 7(1), 5054.Sharp C., DuPont B., 2016. A multi-objective genetic algorithm method for wave energy converter array optimization.Proc. of 35th ASME 2016, Busan, Korea

parameter optimization in wave energy design by a genetic...

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